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Mirrors > Home > MPE Home > Th. List > ereldm | Structured version Visualization version GIF version |
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereldm.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ereldm.2 | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Ref | Expression |
---|---|
ereldm | ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ereldm.2 | . . . 4 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
2 | 1 | neeq1d 2995 | . . 3 ⊢ (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅)) |
3 | ecdmn0 8766 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | |
4 | ecdmn0 8766 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) |
6 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
7 | erdm 8728 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 8 | eleq2d 2814 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) |
10 | 8 | eleq2d 2814 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) |
11 | 5, 9, 10 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 dom cdm 5672 Er wer 8715 [cec 8716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-er 8718 df-ec 8720 |
This theorem is referenced by: erth 8768 brecop 8820 eceqoveq 8832 |
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